smoldering connectors RSV 1000R 2001

Hi,
I own this bike for a week and loved it till now.
I had the following problem:
Trouble to start my RSV 1000R 2001, thought maybe the charging system or bad battery.
Did let bike sit for a few hours and started right away.
Rode for about 10km, fuel light went on, few hundred meters later it just died.

A long walk and pick up of bike later I wanted to take battery out to recharge and noticed a smoldered connector to the part that charges the battery with a consistent current (sorry, whats the name for it?).
The battery is a Motobat and marked as 2011.
The bike had a Dynojet fitted and I am not sure how professional that was done, don't even know what exactly it does yet or how to change settings.

I did get the manual for the RSV and started reading on this Forum.
There were things like the connecters on Aprilia are a known failure but I just want to make sure I am not chasing the wrong thing.